Singularity Functions for Bending Moment

Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a particular function.

Another cut at a point between M and B allows for the bending moment for the segment from midpoint M to the end of the beam to be described by a different function. The key to simplifying the representation is combining these functions into a single expression that adapts based on the position along the beam.

Where w₀ is the distributed load applied over the length from M to the end of the beam. The expression is formed by including the second function in calculations only for positions beyond midpoint M, effectively using a conditional approach to manage the discontinuity. Furthermore, the distribution of the load along the beam, and the resulting shear force, can also be depicted using singularity functions. This method, often employing Macaulay's brackets for representation, streamlines the calculation of bending moments in beams with varying loading conditions.